SUMMARY
It is shown how an electronic analog computer can be utilized in the solution of boundary value problems for ordinary differential equations by the method of super-position. The procedure is illustrated by application to two beam problems. Some of the attendant advantages and disadvantages are discussed.
1. Introduction
-Frequently, boundary value problems for ordinary differential equations can be solved with the aid of an electronic analog computer. Since the computer is a device which naturally solves initial value problems, the 3V3 problem must first be reduced to a sequence of IV problems. The simplest procedure for doing this is one of trial and error. It consists of generatin~
solutions which satisfy the prescribed conditions at one end point, and varying the remaining "init,ial"
values and parameters, until the resulting solution also satisfies the prescribed conditions at the other beforehand. However, an important advantage of the method is that it can be used equally well with both linear and nonlinear problems.
Another means of obtaining a solution to a BV prob-lem by solving IV problems is based on the principle of superposition. It is, of course, applicable only to linear problems, and consists of finding the correct linear combination of solutions to the DE which will satisfy all the prescribed boundary conditions. The method is ideal when a fundamental set of solutions computer in conjunction with the method of superposi-tion lies in the facility with which it can be used to generate the required solutions.
This work was carried out at Project Cyclone, Reeves Instrument Corporation, under V. S. Navy, Bureau of Aeronautics Contract NOas-54-545-c; and was present-ed at the meeting of the Association for COlllputing and DE-differential equation.
26 advan-tages and disadvanadvan-tages of the method.
2. The Boundary Value Problem par.entheses refer to the bibliography).
5 A.N. Krylov (4), and others have used this method, obtaining the solutions by numerical integration.
Boundary Value Problems
amplitude of vibration in torsion square of the circular frequen-cy of vibration.
The eigenvalues and eigenfunctions of this prob-lem can be found by successive approximations, in the following manner: until the required accuracy is obtained.
After the eigenvalue is located, the condition, above requires the computation of several 6 determin-ants of order n. Consequently it does not seem to be pro-gramming is straightforward. Several problems of this type, but with constant coefficients (i.e. for uniform computa-tions were generally known to four significant figures.
For the problems solved on the computer the eigenvalues were found to be in error by less than 0.3%. The solutions obtained after calculating the correct initial values usually satisfied the prescribed BV to within .01 or .02 volts. Of course, less favorable results might be expected in the case of DE's with variable coefficients. Several sample solutions are shown in Figure 3. They represent the plotting board which have been attained.
Computers and Automation
For this and other 6th order EV problems, it was found that anywhere from 6 to 11 trial values of 11 were re-quired. The total time need~d to obtain each of these eigenvalues was"about an hour.
5. Concluding Remarks
Experience with the method indicates that it can be quite useful in many problems. Among its good features are:
1. It is relatively fast and accurate. The program-ming and the computer setup are both usually simple, and only a small amount of computing equipment is needed.
') It yields not only the solution to a problem, but also its derivatives up to (n-l)th order (for a DE of order n). For example, in the problem for the bending of a beam, one can obtain the deflection, slope, bend-ing moment, and shear force, and all with approximate-ly equal accuracy. This is frequentapproximate-ly not the case in other methods. For example, in variational methods, like the Rayleigh-Ritz method, the accuracy of the higher derivatives JIlay be very bad. In fact, the higher derivatives of the successive approximations may not converge altogether.
3. The method may be especially valuable for prob-lems with DE's which contain several powers of A, e.g.
the DE for the buckling of a twisted shaft,
auiv+~ull =-2Au" _ £ u .
a a
The application of finite difference methods to such problems leads to matrices with elements which are polynomials in A, and these are not as easy to solve as the usual matrix eigenvalue problems.
4. The method is applicable to DE's which are not self-adjoint (provided A is rea!).
The most serious di sadvantages in the application of the method are:
1. In some problems only the first few eigenvalues can be obtained. This is true, for example, in beam vibration problems. The reason for this is that' the solution to the DE contain terms which growexponenti-ally, and these increase with A9. Sometimes it is not too difficult to remedy this situation by using the transformation u = e BXV , and solving for v. The con-stant a is adjusted so that v does not grow too rapidly.
Of course, there are also many problems in applied mechanics where this difficulty does not occur, e.g.
problems in the stability of circular arches and of beams on an elastic foundation.
2. Accurate function generation may be difficult sometimes. This probably represents the main source of error in problems for DE's with variable coefficients.
Frequently, however, the magnitude of the error can be quickly ascertained. For example, upper and lower bounds for the frequencies of vibration of a beam can be easily obtained by approximating EIb(x) above and m(x) below and vice versa.
9 See 5.
BIBLIOGRAPHY
1. "Application of the Electronic Differential Analyzer to EigenvalueProblems", by G. Corcos, T. Howe, U. Rauch and J. Sellers, Project Cyclone Symposium II, part 2, 1952, pp. 17-24.
2. "Application of the Electronic Differential Analyzer to Oscillation of Beams, including' Shear and Rotatory' Inertia", by C. Howe and R. Howe, Journal of Applied
Mechanics, v. 22, 1955, pp. 13-19.
3. "The Evaluation of Transient Temperature Distribution ih a Dielectric in an Alternating Field", by C. Copple, D. R. Hartree, A. Porter and H. Tyson, Journal df the Institution of Electrical Engineers, v. 85, 1939, pp. 56-66.
4. Collected Works of A. N. Krylov (Russian), Akademyia Nauk USSR, Moscow-Leningrad, 1937, pp. 311 ff.
5. "NUmerical Analysis", by D. R. Hartree, Oxford Uni·
versity Press, London, 1952, pp. 143-144.
~---;+
.... ---~ +
+26V A HOLD RELAY
NC
I
MII = for
E Ibu"= M seale constants
Figure 1. Schematic, beam bending problem 28
-Boundary Value Problems
I
-k!m>---~---ym---~---~---~
~---.~----~<
Mit: X (mu+S9) T': X (Su+18)
8'"..1-GJ
Cj or. scale constants
Figure 2. Schematic, coupled beam vibrations
FIRST MODE
Figure 3. Coupled modes o( vibration
kzm
x
FOl.m'H MODE
x
(=ont'd on page 39)